Maximum Principles and Boundary Value Problems for First Order Neutral Functional Differential Equations

Volume 2009, Article ID 141959,26pages doi:10.1155/2009/141959

Research Article

Maximum Principles and Boundary Value

Problems for First-Order Neutral Functional

Differential Equations

Alexander Domoshnitsky, Abraham Maghakyan,

and Roman Shklyar

Department of Mathematics and Computer Science, Ariel University Center of Samaria, Ariel 44837, Israel

Correspondence should be addressed to Alexander Domoshnitsky,[email protected]

Received 11 April 2009; Revised 25 August 2009; Accepted 3 September 2009

Recommended by Marta A D Garc´ıa-Huidobro

We obtain the maximum principles for the first-order neutral functional differential equation

Mxt ≡ xt−Sxt−Axt Bxt ft, t ∈ 0, ω,whereA : C0,ω → L∞0,ω, B :

C0,ω → L∞_0,ω, andS : L∞_0,ω → L∞_0,ω are linear continuous operators,AandB are positive operators,C0,ωis the space of continuous functions, andL∞0,ωis the space of essentially bounded

functions defined on0, ω. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.

Copyrightq2009 Alexander Domoshnitsky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Preliminary

This paper is devoted to the maximum principles and their applications for first order neutral functional differential equation.

Mxt≡xt−Sxt−Axt Bxt ft, t∈0, ω, 1.1

whereA:C0,ω → L∞_0,ω, B:C0,ω → L∞_0,ω, and S:L∞0,ω → L∞_0,ωare linear continuous Volterra operators, the spectral radiusρSof the operatorS is less than one, C0,ω is the space of continuous functions,L∞_0,ωis the space of essentially bounded functions defined on

0, ω. We consider1.1with the following boundary condition:

where l : D0,ω → R1 is a linear bounded functional defined on the space of absolutely continuous functionsD_0,ω. By solutions of1.1we mean functionsx:0, ω → R1_{from the} spaceD0,ωwhich satisfy this equation almost everywhere in0, ωand such thatx∈L∞_0,ω.

We mean the Volterra operators according to the classical Tikhonov’s definition.

Definition 1.1. An operatorT is called Volterra if any two functionsx1andx2coinciding on an interval0, ahave the equal images on0, a, that is,Tx1t Tx2tfort∈0, aand for each 0< a≤ω.

Maximum principles present one of the classical parts of the qualitative theory of ordinary and partial differential equations 1. Although in many cases, speaking about maximum principles, authors mean quite different definitions of maximum principlessuch as e.g., corresponding inequalities, boundedness of solutions and maximum boundaries principles, there exists a deep internal connection between these definitions. This connection was discussed, for example, in the recent paper2. Main results of our paper are based on the maximum boundaries principle, that is, on the fact that the maximal and minimal values of the solution can be achieved only at the points 0 orω.The boundaries maximum principle in the case of the zero operatorSwas considered in the recent papers2,3. In this paper we develop the maximum boundaries principle for neutral functional differential equation

1.1and on this basis we obtain results on existence and uniqueness of solutions of various boundary value problems.

Although several assertions were presented as the maximum principles for delay differential equations, they can be only interpreted in a corresponding sense as analogs of classical ones for ordinary differential equations and do not imply important corollaries, reached on the basis of the finite-dimensional fundamental systems. For example, results, associated with the maximum principles in contrast with the cases of ordinary and even partial differential equations, do not add so much in problems of existence and uniqueness for boundary value problems with delay differential equations. The Azbelev’s definition of the homogeneous delay differential equation4,5allowed his followers to consider questions of existence, uniqueness and positivity of solutions on this basis. The first results about the maximum principles for functional differential equations, which were based on the idea of the finite-dimensional fundamental system, were presented in the paper2.

Neutral functional differential equations have their own history. Equations in the form

xt−qtxτt m

i1

bitxhit ft, t∈0,∞, 1.3

were considered in the known books 6–8 see also the bibliography therein, where existence and uniqueness of solutions and especially stability and oscillation results for these equations were obtained. There exist problems in applications whose models can be written in the form9

xt−qtxτt m

i1

bitxhit ft, t∈0,∞. 1.4

Let us note here that the operatorS :L∞_0,∞ → L∞_0,∞in1.1can be, for example, of the following forms:

Syt m j1qjty

τjt

, whereτjt≤t, y

τjt

0 ifτjt<0, t∈0,∞, 1.5

or

Syt n

i1

_t

0

kit, sysds, t∈0,∞, 1.6

whereqjtare essentially bounded measurable functions,τjtare measurable functions for j1, . . . , m,andkit, sare summable with respect tosand measurable essentially bounded with respect totfori1, . . . , n. All linear combinations of operators1.5and1.6and their superpositions are also allowed.

The study of the neutral functional differential equations is essentially based on the questions of the action and estimates of the spectral radii of the operators in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions. Operator 1.5, which is a linear combination of the internal superposition operators, is a key object in this topic. Properties of this operator were studied by Drakhlin

10,11. In order to achieve the action of operator1.5in the space of essentially bounded functionsL∞_0,∞, we have for eachjto assume that mes{t:τjt c}0 for every constantc. Let us suppose everywhere below that this condition is fulfilled. It is known that the spectral radius of the integral operator1.6,considered on every finite intervalt∈0, ω,is equal to zerosee, e.g.,4. Concerning the operator1.5, we can note the sufficient conditions of the fact that its spectral radiusρSis less than one.Define the setκjε{t∈0,∞: t−τjt≤ε} and κε

_m

j1κjε. If there exists such ε that mes κε 0, then on every finite interval t ∈ 0, ω the spectral radius of the operatorSdefined by the formula1.5fort ∈ 0, ω is zero. In the case mesκε > 0,the spectral radius of the operatorS defined by1.5on the finite interval t ∈ 0, ω is less than one if ess sup_t∈κ

ε

_m

j1|qjt| < 1. The inequality ess sup_t∈0,∞m_j1|qjt|<1 implies that the spectral radiusρSof the operatorSconsidered on the semiaxist∈0,∞and defined by1.5,satisfies the inequalityρS<1.Usually we will also assume that τj are nondecreasing functions forj 1, . . . , m.

Various results on existence and uniqueness of boundary value problems for this equation and its stability were obtained in 4, where also the basic results about the representation of solutions were presented. Note also in this connection the papers in12–

15, where results on nonoscillation and positivity of Green’s functions for neutral functional differential equations were obtained.

It is known4that the general solution of1.1has the representation

xt

_t

0

Ct, sfsdsXtx0, 1.7

basis of representation1.7, the results about differential inequalitiesunder corresponding conditions, solutions of inequalities are greater or less than solution of the equation can be formulated in the following form of positivity of the Cauchy function Ct, s and the solutionXt. Results about comparison of solutions for delay differential equations solved with respect to the derivativei.e., in the case whenSis the zero operatorwere obtained in

2,15,16, where assertions on existence and uniqueness of solutions of various boundary value problems for first order functional differential equations were obtained.

All results presented in the paper 15 and in the book 16 for equation with the difference of two positive operators are based on corresponding analogs of the following assertion15: Let the operatorAand the Cauchy functionCt, sof equation

Mxt≡xt−Sxt Bxt ft, t∈0, ω, 1.8

be positive for0≤s≤t≤ω, then the Cauchy functionCt, sof1.1is also positive for0≤s≤t≤ ω.

This result was extent on various boundary value problems in16in the form: Let the operatorAand Green’s functionGt, sof problem1.8,1.2be positive in the square0, ω×0, ω and the spectral radius of the operatorΩ:C0,ω → C0,ωdefined by the equality

Ωxt

ω

0

Gt, sAxsds, 1.9

be less than one, then Green’s functionGt, sof problem1.1,1.2is positive in the square0, ω×

0, ω.

The scheme of the proof was based on the reduction of problem1.8,1.2withc0, to the equivalent integral equationxt Ωxt ϕt,whereϕt ω₀Gt, sfsds.It is clear that the operatorΩ is positive if the operator A and the Green’s functionGt, s are positive. If the spectral radiusρΩor, more roughly, the normΩof the operatorΩ : C_0,ω → C_0,ωare less than one, then there exists the inverse bounded operatorI−Ω−1

I Ω Ω2_{· · · : C}

0,ω → C0,ω,which is of course positive. This implies the positivity of the Green’s functionGt, sof problem1.1,1.2. In order to get the inequalityρΩ<1, the classical theorems about estimates of the spectral radius of the operatorΩ:C_0,ω → C_0,ω

17can be used. All these theorems are based on a corresponding “smallness” of the operator

Ω,which is actually close to the conditionΩ< 1.In order to get positivity ofCt, sand Gt, sa corresponding smallness ofBwas assumed.

Below we present another approach to this problem starting with the following question: how can one conclude about positivity of Green’s functionGt, sin the cases when the spectral radius satisfies the opposite inequality ρΩ ≥ 1 or Green’s function Gt, s changes its sign? Note, that in the case, when the operatorS : L∞0,ω → L∞_0,ωis positive and its spectral radius is less than one, the positivity of the Cauchy functionCt, sof1.8

We assume that the spectral radius of the operatorS:L∞0,ω → L∞_0,ωis less than one. In this case we can rewrite1.1in the equivalent form

Nxt≡xt−I−S−1A−Bxt I−S−1ft, t∈0, ω, 1.10

and its general solution can be written in the form

xt

t

0

C0t, sI−S−1fsdsXtx0, 1.11

whereC0t, sis the Cauchy function of1.10 4. Note that this approach in the study of neutral equations was first used in the paper14. Below in the paper we use the fact that the Cauchy functionC0t, scoincides with the fundamental function of1.10. It is also clear that

t

0

Ct, sfsds

t

0

C0t, sI−S−1fsds. 1.12

2. About Maximum Boundaries Principles in the Case of

Difference of Two Positive Volterra Operators

In this paragraph we consider the equation

Mxt≡xt−Sxt−Axt Bxt ft, t∈0,∞, 2.1

whereA : C0,∞ → L_0,∞_∞, B : C0,∞ → L∞_0,∞ andS : L_0,∞_∞ → L∞_0,∞are positive linear continuous Volterra operators and the spectral radiusρSof the operatorSis less than one. These operatorsAandBareu-bounded operators and according to18, they can be written in the form of the Stieltjes integrals

Axt

t

0

xξdξat, ξ, Bxt

t

0

xξdξbt, ξ, t∈0,∞, 2.2

respectively, where the functionsa·, ξ and b·, ξ : 0, ω → R1 _{are measurable for ξ ∈}

0, ω, at,·andbt,·:0, ω → R1_{has the bounded variation for almost allt∈0, ωand} t

ξ0at, ξ, tξ0bt, ξare essentially bounded.

Consider for convenience2.1in the following form:

Mxt≡xt−Sxt−

_t

0

xξdξat, ξ

_t

0

We can study properties of solution of2.3on each finite interval0, ωsince every solution xtof2.1satisfies also the equation

Mxt≡xt−Sxt−

_t

0

xξdξat, ξ

_t

0

xξdξbt, ξ ft, t∈0, ω. 2.4

Consider also the homogeneous equation

Mxt≡xt−Sxt−Axt Bxt 0, t∈0,∞, 2.5

and the following auxiliary equations which are analogs of the so-called s-trancated equations defined first in5

Msxt≡xt−

Ssx

t−Asxt Bsxt 0, t∈s,∞, s≥0, 2.6

where the operatorsAs:Cs,∞ → L∞_s,∞andBs:Cs,∞ → L∞_s,∞are defined by the formulas

Asxt

_t

s

xξdξat, ξ, Bsxt

_t

s

xξdξbt, ξ, t∈s,∞, 2.7

and the operatorSs : L∞_s,∞ → L∞_s,∞ is defined by the equality Ssyst Syt,where yst ytfort≥sandyt 0 fort < s.We have

Ssy

t

m

j1 qjty

τjt

, where τjt≤t, y

τjt

0 ifτjt< s, t∈s,∞, 2.8

for the operator described by formula1.5, and

Ssy

t

n

i1

t

s

kit, sysds, t∈s,∞, 2.9

for the operator described by formula1.6. It is clear thatρSs< 1 for everys∈0,∞if ρS<1.

Functions from the space D_s,∞ of absolutely continuous functions x : s,∞ → R1_{, x∈L}∞

s,∞,satisfy2.6almost everywhere ins,∞, we call solutions of this equation. It was noted above that the general solution of2.1has the representation4

xt

_t

0

where the function Ct, s is the Cauchy function of 2.1.We use also formula 1.12

connectingCt, sand the Cauchy functionC0t, sof1.10. Note thatC0t, sis a solution of

2.6as a function of the first argumenttfor every fixedsand satisfies also the equation

Nsxt≡xt I−Ss−1−Asxt Bsxt 0, t∈s,∞. 2.11

Let us formulate our results about positivity of the Cauchy functionCt, sand the maximum boundaries principle in the case when the conditionCt, s>0 is not assumed.

Consider the equation

xt−Sxt−

_g2t

g1t

xξdξat, ξ

_h2t

h1t

xξdξbt, ξ ft, t∈0,∞. 2.12

Theorem 2.1. LetS:L∞_0,∞ → L∞_0,∞be a positive Volterra operator,ρS<1, 0≤h1t≤h2t≤ g1t≤g2t≤t,let the functionsat, ξandbt, ξbe nondecreasing functions with respect toξfor almost everyt, and let the following inequality be fulfilled:

h2t

ξh1t

bt, ξ≤ g2t

ξg1t

at, ξ, t∈0,∞, 2.13

then the Cauchy function Ct, s of 2.12 and its derivative satisfy the inequalities Ct, s > 0, C_tt, s≥0for0≤s≤t <∞.

Consider now the equation

xt−Sxt m

i1

−

_g2it

g1it

xξdξait, ξ

_h2it

h1it

xξdξbit, ξ

ft, t∈0,∞.

2.14

Theorem 2.2. Let S : L∞_0,∞ → L∞_0,∞ be a positive Volterra operator,ρS < 1, 0 ≤ h1it ≤ h2it ≤ g1it ≤ g2it ≤ t,let the functionsait, ξand bit, ξbe nondecreasing functions with respect toξ for almost everytand let the following inequalities

h2it

ξh1it

bit, ξ≤ g2it

ξg1it

ait, ξ, t∈0,∞, i1, . . . , m, 2.15

be fulfilled, then the Cauchy functionCt, sof2.14and its derivativeC_tt, ssatisfy the inequalities Ct, s>0andC_tt, s≥0for0≤s≤t <∞.

Consider the delay equation

xt−Sxt− m

i1 aitx

git

m

i1

where

xξ 0 for ξ <0, 2.17

with ai, bi ∈ L∞_0,∞ and measurable functions gi and hi i 1, . . . , m. This equation is a particular case of2.14.

Theorem 2.3. LetS :L∞_0,∞ → L∞_0,∞ be a positive Volterra operator,ρS< 1, hit≤ git≤ t and0 ≤bit≤aitfort∈0,∞, i1, . . . , m,then the Cauchy functionCt, sof 2.16and its derivativeC_tt, ssatisfy the inequalitiesCt, s>0andC_tt, s≥0for0≤s≤t <∞.

Example 2.4. The inequality on deviating argumenthit ≤gitis essential as the following equation

xt−x0 b1txh1t 0, t∈0,∞, 2.18

demonstrates. This is a particular case of2.16, whereSis the zero operator,m1, g1t≡ 0, a1t≡1,

h1t

⎧ ⎨ ⎩

0, 0≤t <2,

2, 2≤t,

b1t

⎧ ⎪ ⎨ ⎪ ⎩

0, 0≤t <2, 1

2, 2≤t.

2.19

The function

xt≡Ct,0

⎧ ⎪ ⎨ ⎪ ⎩

t1, 0≤t <2,

4−1

2t, 2≤t,

2.20

is a nontrivial solution of2.18and its Cauchy functionCt, ssatisfies the equality

Ct, s

⎧ ⎪ ⎨ ⎪ ⎩

1, 0< s≤2, 0≤t <2,

1−1

2t−2, 0< s≤2, 2≤t,

2.21

Consider the integrodifferential equation

xt−Sxt− m

i1

_g2it

g1it

mit, ξxξdξ m

i1

_h2it

h1it

kit, ξxξdξft, t∈0,∞,

xξ 0 for ξ <0,

2.22

as a particular case of2.14.

Let us define the functionsh0_jit max{0, hjit}and gji0t max{0, gjit},where j1,2.

Theorem 2.5. LetS :L∞_0,∞ → L∞_0,∞be a positive Volterra operator,ρS<1, h1it≤h2it ≤ g1it ≤ g2it ≤ t, kit, ξ ≥ 0, mit, ξ ≥ 0fort, ξ ∈ 0,∞, i 1, . . . , m,and the following inequalities be fulfilled:

_h0 2it

h0 1it

kit, ξdξ≤

g0 2it

g0 1it

mit, ξdξ, t∈0,∞, i1, . . . , m, 2.23

then the Cauchy function of 2.22and its derivativeC_tt, ssatisfy the inequalitiesCt, s>0and C_tt, s≥0for0≤s≤t <∞.

Consider the equation

xt−Sxt−

g2t

g1t

mt, ξxξdξbtxht ft, t∈0,∞.

xξ 0 forξ <0.

2.24

Let us define

χt, s

⎧ ⎨ ⎩

0, t < s,

1, t≥s.

2.25

In the following assertion the integral term is dominant.

Theorem 2.6. LetS : L∞_0,∞ → L∞_0,∞ be a positive Volterra operator,ρS < 1, ht ≤ g1t ≤ g2t≤t, mt, ξ≥0, bt≥0fort, ξ∈0,∞and the following inequalities be fulfilled:

btχht,0≤

g0 2t

g0 1t

mt, ξdξ, t∈0,∞, 2.26

Consider the equation

xt−Sxt−atxgt

_h2t

h1t

kt, ξxξdξft, t∈0,∞,

xξ 0 forξ <0.

2.27

In the following assertion the termatxgtis a dominant one.

Theorem 2.7. LetS:L∞_0,∞ → L∞_0,∞be a positive Volterra operator,ρS<1, kt, ξ≥0, h1t≤ h2t≤gt≤t, at≥0fort, ξ∈0,∞and the following inequalities be fulfilled:

h0 2t

h0 1t

kt, ξdξ≤at, t∈0,∞, 2.28

then the Cauchy function of 2.27and its derivativeC_tt, ssatisfy the inequalitiesCt, s>0and C_tt, s≥0for0≤s≤t <∞.

Consider now the equation

xt−Sxt−atxgtbtxht−

g2t

g1t

mt, ξxξdξ

h2t

h1t

kt, ξxξdξ

ft, t∈0,∞,

xξ 0 forξ <0.

2.29

In the following assertion we do not assume inequalitieskt, ξ≤mt, ξorbt≤at. Here the sumatxgt g2t

g1tmt, ξxξdξis a dominant term.

Theorem 2.8. LetS : L∞_0,∞ → L∞_0,∞ be a positive Volterra operator,ρS < 1, ht ≤ g1t ≤ g2t≤ t, h1t≤h2t≤gt≤ t, kt, ξ ≥0, mt, ξ≥ 0, at≥0, bt≥0fort, ξ ∈0,∞ and let the following inequalities be fulfilled:

btχht,0≤

g0 2t

g0 1t

mt, ξdξ,

h0 2t

h0 1t

kt, ξdξ≤at,

2.30

The proofs of Theorems2.1–2.8are based on the following auxiliary lemmas.

Lemma 2.92. LetSbe the zero operator. Then the following two assertions are equivalent:

1for every positivesthere exists a positive functionvs∈Ds,∞such thatMsvst≤0for t∈s,∞,

2the Cauchy functionCt, sof2.1is positive for0≤s≤t <∞.

Lemma 2.10. LetS:L∞_0,∞ → L∞_0,∞be a positive Volterra operator,ρS<1and for every positive sthere exist a positive functionvs∈Ds,∞such thatMsvst≤0fort∈s,∞,then the Cauchy functionC0t, sof 1.10is positive for0≤s≤t <∞.

Proof. Using the conditionρS < 1,we can write1.1in the form 1.10. The positivity of the operator S : L∞_0,∞ → L∞_0,∞ implies that the inequalityI −S−1f ≥ 0 follows from the inequality f ≥ 0.It is clear that the inequality Msvst ≤ 0 fort ∈ s,∞ implies the inequalityNsvst ≤ 0 fort ∈ s,∞.Now according toLemma 2.9, we obtain that C0t, s>0 for 0≤s≤t <∞.

Lemma 2.11. LetA :C0,∞ → L∞_0,∞, B :C0,∞ → L∞_0,∞andS: L_0,∞_∞ → L∞_0,∞be positive Volterra operators,ρS<1and for everys∈0,∞the inequality

As1t≥Bs1t fort∈s,∞, 2.31

be fulfilled. ThenC0t, s>0for0≤s≤t <∞.

In order to proveLemma 2.11we setvst≡1, t∈s,∞for everys∈0,∞in the assertion 1 ofLemma 2.10.

Remark 2.12. The condition

A1t≥B1t fort∈0, ω 2.32

cannot be set instead of condition 2.31 in Lemma 2.11 as Example 2.4 demonstrates. It is clear that this inequality is fulfilled for 2.18, where Axt x0 and Bxt b1txh1t,the functionsh1andb1are defined by formula2.19respectively. The operator Asis the zero one for everys >0 and consequentlyAs1t 0 fort∈ s,∞, B21t 1/2 fort∈2,∞and condition2.31is not fulfilled fors2.

Lemma 2.13. Let A : C0,∞ → L∞_0,∞, B : C0,∞ → L∞_0,∞ and S : L_0,∞_∞ → L∞_0,∞ be positive Volterra operators,ρS < 1and inequality2.31be fulfilled for everys ∈ 0,∞.Then Ct, s>0, ∂/∂tC0t, s≥0and (∂/∂tCt, s≥0for0≤s≤t <∞.

Proof. According toLemma 2.11, we haveC0t, s>0 for 0≤s≤t <∞.The positivity of the operatorS : L∞_0,∞ → L∞_0,∞ and formula 1.12implies now thatCt, s ≥ C0t, s > 0 for 0≤s≤t <∞.

The following integral equation

xt

t

s

∞

n0

Sn_sAs−Bsxξdξ1 2.33

is equivalent to2.11with the conditionxs 1.

The spectral radius of the operatorT:Cs,ω → Cs,ω,defined by the equality

Txt

t

s

∞

n0

Sn_sAs−Bsxξdξ, t∈s, ω, 2.34

is zero for every positive numberω4. Let us build the sequence

xm1t

_t

s

∞

n0

Sn_sAs−Bsxmξdξ1, 2.35

where the iterations start with the constantx0t≡1 fort∈s, ω.

The sequence of functionsxmtconverges in the spaceCs,ω to the unique solution xtof2.33on the intervals, ω. It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solutionytof2.11with the initial conditionys 1 and the solutionxtof2.33coincide fort∈s, ω.

Positivity of the operatorS, the inequalitiesρS <1 and2.31imply nonnegativity of the derivatives

x_m1t

∞

n0

Sn

sAs−Bsxmt, t∈s, ω. 2.36

Let us prove now that the sequence xm of nondecreasing functions converges to the nondecreasing functionx.Assume in the contrary that there exist two pointst1 < t2,such thatxt1> xt2.Let us chooseε <xt1−xt2/2.There exists a numberN1εsuch that |xt1−xmt1| < εform ≥ N1ε, and there existsN2εsuch that|xmt2−xt2| < ε for m≥N2ε.It is clear thatxmt1> xmt2form≥max{N1ε, N2ε}.This contradicts to the fact thatxmtnondecreases.

We have proven that for every positiveω,the solutionxof2.33is nondecreasing for t ∈ s, ω. It means that the solutionxof2.11is nondecreasing for everyt ∈ s,∞and consequently∂/∂tC0t, s≥0 for 0≤s≤t <∞.

Positivity of the operatorS,the inequalityρS<1 and formula1.12imply now the inequality∂/∂tCt, s≥0 for 0≤s≤t <∞.

To prove Theorems2.1–2.8it is sufficient to note that the conditions of each theorem imply inequality2.31.

Remark 2.14. The space of solutions of the homogeneous equation

in the caseρS<1 is one dimensional. All nontrivial solutions of2.37are proportional to C0t,0.One of the assertions ofLemma 2.11claims that∂/∂tC0t,0≥0 for 0≤t <∞,that is, all nontrivial positive solutions do not decrease. This allows us to considerLemma 2.13

as the maximum boundaries principle for 2.1. Theorems 2.1–2.8 present the sufficient conditions of this maximum principle for the equations2.12,2.14,2.16,2.22,2.24,

2.27and2.29respectively.

Remark 2.15. The conditionρS < 1 about the spectral radius of the operatorS : L∞_0,∞ → L∞_0,∞is essential as the following example demonstrates.

Example 2.16. Consider the equation

xt−x

t 2

ft, t∈0,∞. 2.38

The spectral radius of the operator S : L∞_0,∞ → L∞_0,∞, defined by the formulaSyt yt/2,is equal to one. All other conditions of Theorems 2.1–2.8for the zero operators A andBare fulfilled. The space of solutions of this neutral homogeneous equation is infinitely dimensional. Every linear functionsx1−ctsatisfy the homogeneous equation

xt−x

t 2

0, t∈0,∞. 2.39

Ifc >0, the solutionsxare decreasing.

3. About Nondecreasing Solutions of Neutral Equations

Let us consider the equation

Mxt≡xt Sxt−Axt Bxt ft, t∈0,∞, 3.1

whereA : C0,∞ → L∞_0,∞ andB : C0,∞ → L∞_0,∞ are positive linear continuous Volterra operators, and the spectral radiusρSof the operatorS : L_0,∞_∞ → L∞_0,∞ is less than one. If the operatorSis positive, thenIS−1 I−SS2_−S3_{· · · is not generally speaking a} positive operator. This is the main difficulty in the study of positivity of the solutionxand its derivativex. All previous results about the positivity of solutions for this equation assumed the negativity of the operatorSsee, e.g.,12,13,15. In this paragraph we propose results about positivity of solutions in the case of the positive operatorSdefined by the equality

Syt qtyrt, wherert≤t, yrt 0 ifrt<0, t∈0,∞, 3.2

Let us start with the equation

xt qtxrt−atxgtbtxht ft, t∈0,∞,

Theorem 3.1. Assume that the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞defined by equality3.2is less than one,rt, htandgtare nondecreasing functions, and the coefficients satisfy the inequalities at ≥ 0, bt ≥ 0, qt ≥ 0, gt ≥ ht and at−btχht,0− qtartχgrt,0≥0fort∈0,∞,then the solutionxof the equation

xt qtxrt−atxgtbtxht 0, t∈0,∞,

xξ xξ 0 forξ <0, 3.4

such thatx0>0,satisfies the inequalitiesxt≥0, xt≥0fort∈0,∞and in the case, when there existsεsuch that0≤qt≤ε <1,the solutionxof3.3is nonnegative and nondecreasing for every positive nondecreasing functionf∈L∞_0,∞.

Consider the equation

xt qtxrt m

i1

−

g2it

g1it

xξdξait, ξ

h2it

h1it

xξdξbit, ξ

ft, t∈0,∞,

xξ 0 for ξ <0.

3.5

Theorem 3.2. Let the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞ defined by equality

3.2 be less than one, rt be a nondecreasing function and the functions ait, ξ and bit, ξ be nondecreasing functions with respect toξ, 0 ≤ h1it ≤ h2it ≤ g1it ≤ g2it ≤ t, qt ≥ 0, and the following inequalities be fulfilled

h2it

ξh1it

bit, ξ qtχrt,0 g2irt

ξg1irt

airt, ξ≤ g2it

ξg1it

ait, ξ, 3.6

fort∈0,∞, i1, . . . , m,then the solutionxof the equation

xt qtxrt m

i1

−

g2it

g1it

xξdξait, ξ

h2it

h1it

xξdξbit, ξ

0, t∈0,∞,

xξ 0 forξ <0.

3.7

Consider the integrodifferential equation

xt qtxrt− m

i1

_g2it

g1it

mit, ξxξdξ m

i1

_h2it

h1it

kit, ξxξdξft, t∈0,∞,

xξ xξ 0 forξ <0,

3.8

Theorem 3.3. Let the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞ defined by equality

3.2be less than one, rt be a nondecreasing function and h1it ≤ h2it ≤ g1it ≤ g2it ≤ t, qt≥0,kit, ξ≥0, mit, ξ≥0fort, ξ∈0,∞,and the following inequalities be fulfilled

_h2it

h1it

kit, ξdξqtχrt,0

_g2irt

g1irt

mit, ξdξ≤

_g2it

g1it

mit, ξdξ, 3.9

t∈0,∞, i1, . . . , m,then the solutionxof the equation

xt qtxrt− m

i1

g2it

g1it

mit, ξxξdξ m

i1

h2it

h1it

kit, ξxξdξ0, t∈0,∞,

xξ xξ 0 forξ <0,

3.10

such thatx0>0,satisfies the inequalitiesxt≥0, xt≥0fort∈0,∞and in the case, when there existsεsuch that0≤qt≤ε <1,the solutionxof3.8is nonnegative and nondecreasing for every positive nondecreasing functionf∈L∞_0,∞.

Consider the equation

xt qtxrt−

_g2t

g1t

mt, ξxξdξbtxht ft, t∈0,∞,

xξ xξ 0 forξ <0.

3.11

In the following assertion the integral term is dominant.

Theorem 3.4. Let the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞ defined by equality

3.2be less than one,rtbe a nondecreasing function,qt ≥ 0, bt ≥ 0, mt, ξ ≥ 0, ht ≤ g1t≤g2t≤tfort, ξ∈0,∞,and the following inequality be fulfilled

btχht,0 qtχrt,0

g2rt

g1rt

mt, ξdξ≤

g2t

g1t

then the solutionxof the equation

xt qtxrt−

_g2t

g1t

mt, ξxξdξbtxht 0, t∈0,∞,

xξ xξ 0 forξ <0,

3.13

such thatx0>0,satisfies the inequalitiesxt≥0, xt≥0fort∈0,∞and in the case, when there existsεsuch that0≤qt≤ε < 1,the solutionxof 3.11is nonnegative and nondecreasing for every positive nondecreasing functionf ∈L∞_0,∞.

Consider the equation

xt qtxrt−atxgt

h2t

h1t

kt, ξxξdξft, t∈0,∞,

xξ xξ 0 forξ <0.

3.14

In the following assertion the termatxgtis dominant.

Theorem 3.5. Let the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞ defined by equality

3.2be less than one,rtbe a nondecreasing function,h1t≤h2t≤gt≤t, qt≥0, kt, ξ≥ 0, at≥0fort, ξ∈0,∞,and the following inequality

_h0 2t

h0 1t

kt, ξdξqtartχrt,0≤at, t∈0,∞, 3.15

be fulfilled, then the solutionxof the equation

xt qtxrt−atxgt

_h2t

h1t

kt, ξxξdξ0, t∈0,∞,

xξ xξ 0 forξ <0.

3.16

such thatx0>0,satisfies the inequalitiesxt≥0, xt≥0fort∈0,∞and in the case, when there existsεsuch that0≤qt≤ε < 1,the solutionxof 3.14is nonnegative and nondecreasing for every positive nondecreasingf∈L∞_0,∞.

Consider now the equation

xt qtxrt−atxgtbtxht−

g2t

g1t

mt, ξxξdξ

_h2t

h1t

kt, ξxξdξft, t∈0,∞,

xξ xξ 0 for ξ <0.

In the following assertion we do not assume inequalitieskt, ξ≤mt, ξorbt≤at. Here the sumatxgt g2t

g1tmt, ξxξdξis a dominant term.

Theorem 3.6. Let the spectral radiusρSof the operatorS :L∞_0,∞ → L∞_0,∞ defined by equality

3.2be less than one,rtbe a nondecreasing function,kt, ξ ≥0, mt, ξ≥ 0, at ≥ 0, bt ≥ 0, qt≥0, ht≤g1t≤g2t≤t, h1t≤h2t≤gt≤tfort, ξ ∈0,∞and the following inequalities be fulfilled

btχht,0 qtχrt,0

_g0 2rt

g0 1rt

mt, ξdξ≤

_g0 2t

g0 1t

mt, ξdξ,

_h0 2t

h0 1t

kt, ξdξqtartχrt,0≤at,

3.18

fort∈0,∞,then the solutionxof the equation

xt qtxrt−atxgtbtxht−

g2t

g1t

mt, ξxξdξ

h2t

h1t

kt, ξxξdξ0, t∈0,∞,

3.19

xξ xξ 0 forξ <0, 3.20

such thatx0>0,satisfies the inequalitiesxt≥0, xt≥0fort∈0,∞and in the case, when there existsεsuch that0≤qt≤ε < 1,the solutionxof 3.17is nonnegative and nondecreasing for every positive nondecreasingf∈L∞_0,∞.

Let us write3.1in the form

ISxt Axt−Bxt ft. 3.21

The spectral radiusρSof the operatorS:L∞_0,∞ → L∞_0,∞is less than one,then there exists the bounded operatorIS−1:L∞_0,∞ → L∞_0,∞and we can write3.21in the form

Nxt≡xt−

∞

n0

−1nSnA−Bxt

∞

n0

−1nSnft. 3.22

Proofs of Theorems3.1–3.6are based on the following auxiliary assertions.

Lemma 3.7. Let A :C_0,∞ → L∞_0,∞, B:C_0,∞ → L∞_0,∞andS:L_0,∞_∞ → L∞_0,∞be positive Volterra operators, the spectral radiusρSof the operatorSbe less than one and

As1t≥Bs1t SsAs1t, t∈s,∞, 3.23

for every nonnegatives.ThenC0t, s>0and∂/∂tC0t, s≥0for0≤s≤t <∞.

Proof. Lemma 2.9is true for 3.22. Let us set vst ≡ 1, t ∈ s,∞ in the assertion 1 of

Lemma 2.9. Condition3.23implies, according toLemma 2.9, thatC0t, s>0 for 0≤s≤t <

∞.

Now let us prove that∂/∂tC0t, s≥0 for 0≤s ≤t <∞.We use the fact that the functionC0t, s,as a function of argumenttfor each fixed positives,satisfies the equation

Nsxt≡xt−

∞

n0

−1nSn_sAs−Bsxt 0, t∈s,∞, 3.24

and the conditionxs 1.

The following integral equation

xt

_t

s

∞

n0

−1nSn_sAs−Bsxξdξ1 3.25

is equivalent to3.24with the conditionxs 1.

The spectral radius of the operatorT:Cs,ω → Cs,ω,defined by the equality

Txt

_t

s

∞

n0

−1nSn_sAs−Bsxξdξ, t∈s, ω, 3.26

is zero for every positive numberω4. Let us build the sequence

x_m1t

t

s

∞

n0

−1nSn

sAs−Bsxmξdξ1, 3.27

where the iterations start with the constantx0t≡1 fort∈s, ω.

The sequence of functionsxmtconverges in the spaceCs,ω to the unique solution xtof3.25on the intervals, ω. It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solutionytof3.24with the initial conditionys 1 and the solutionxtof3.25coincide fort∈s, ω.

Positivity of the operatorS, the inequalitiesρS <1 and3.23imply nonnegativity of the derivatives

x_m1t

∞

n0

Repeating the argumentation used in the proof ofLemma 2.13, we obtain that this sequence of nondecreasing functionsxm converges to the nondecreasing solutionx, that is,

∂/∂tC0t, s≥0 for 0≤s≤t <∞.

Concerning nonhomogeneous3.1we propose the following assertion.

Lemma 3.8. Let A :C0,∞ → L∞_0,∞, B:C0,∞ → L∞_0,∞andS:L_0,∞_∞ → L∞_0,∞be positive Volterra operators, the spectral radiusρSof the operatorSbe less than one and inequality3.23be fulfilled for every nonnegatives.Then the solutionxof the homogeneous equation

Mxt≡xt Sxt−Axt Bxt 0, t∈0,∞, 3.29

such that x0 ≥ 0,satisfies inequalitiesxt ≥ 0, xt ≥ 0 for t ∈ 0,∞.If in addition the nonnegative functionf ∈ L∞_0,∞ satisfies the inequalityft ≥ Sftfort ∈ 0,∞,then the solutionxof 3.1is nonnegative and nondecreasing for every positive nondecreasingf.

Remark 3.9. The inequalityft≥Sftfort∈0,∞is fulfilled if a nonnegative function fis nondecreasing and the norm of the operatorS:L∞_0,∞ → L∞_0,∞is less than one.

Proof ofLemma 3.8.Assertions about nonnegativity of solution x of the homogeneous equationMx 0 and its derivative follows from the equalitiesxt C0t,0 andxt

∂/∂tC0t,0andLemma 3.7. From the representation of solutions of3.22we can write

xt

_t

0 C0t, s

∞

n0

−1nSnfsdsx0C0t,0,

xt ∞

n0

−1nSnft

t

0 ∂

∂tC0t, s ∞

n0

−1nSnfsdsx0∂

∂tC0t,0.

3.30

It is clear now that the inequalityft ≥ Sft for t ∈ 0,∞, positivity of S and nonnegativity of∂/∂tC0t, sfor 0 ≤ s ≤ t < ∞imply the inequalitiesxt ≥ 0, xt ≥ 0 fort∈0,∞.

The proofs of Theorems3.1–3.6follows from the fact that conditions of every theorem imply the conditions ofLemma 3.8for corresponding equations.

Remark 3.10. The condition

A1t≥B1t SA1t, t∈0,∞, 3.31

cannot be set instead of condition3.23asExample 2.4 demonstrates. In this example, the operatorS is the zero one,Axt x0and A1t 1,B1t 0 fort ∈ 0,2and

B1t 1/2 fort≥2,condition3.31fulfilled, the Cauchy functionCt, sof2.18and its derivative change their signs. We noted that the inequality on delays avoids this situation.

Remark 3.11. In the case of the neutral equation

where

ht

⎧ ⎨ ⎩

0, 0≤t <3,

2, 3≤t,

gt

⎧ ⎨ ⎩

0, 0≤t <2,

2, 2≤t,

3.33

the inequalityht≤ gtfort∈0,∞does not avoid the changes of signs ofC0t, sand its derivative:

C0t,2

⎧ ⎨ ⎩

t−1, 2≤t <3,

5−t, 3≤t, 3.34

and∂C0t,2/∂t−1<0 fort >3 andC0t,2<0 fort >5.

This example demonstrates that we cannot set very natural inequality

at−btχht,0−qtartχgrt,0qtbrtχhrt,0≥0, t∈0,∞,

3.35

instead of

at−btχht,0−qtartχgrt,0≥0, t∈0,∞, 3.36

inTheorem 3.1even in the case whenht≤gtfort∈0,∞.

4. Maximum Boundaries Principles in Existence and

Uniqueness of Boundary Value Problems

Consider the boundary value problems of the following type

Mxt≡xt−Sxt−Axt Bxt ft, t∈0, ω, 4.1

lxc, 4.2

wherel:D0,ω → R1is a linear bounded functional andc∈R1.

It was explained inRemark 2.14thatLemma 2.13can be considered as the maximum boundaries principle for 4.1, and Theorems 2.1–2.8 present sufficient conditions of the maximum boundaries principles for equations2.12,2.14,2.16,2.22,2.24,2.27and

The assertions about existence and uniqueness are based on the known Fredholm alternative for functional differential equations.

Lemma 4.14. Let the spectral radius of the operatorS:L_0,∞_∞ → L∞_0,∞be less than one, then boundary value problem4.1,4.2is uniquely solvable for eachf∈L0,ω, c∈R1if and only if the homogeneous problem

Mxt≡xt−Sxt−Axt Bxt 0, t∈0, ω, lx0, 4.3

has only the trivial solution.

Theorem 4.2. LetS:L∞_0,∞ → L∞_0,∞be a positive Volterra operator and its spectral radius satisfy the inequalityρS<1,and for eachs∈0, ωthe inequality

As1t≥Bs1t fort∈s, ω, 4.4

be fulfilled. Then the following assertions are true:

1Ifl : C_0,ω → R1 _{is a linear nonzero positive functional, then boundary value problem}

4.1,4.2is uniquely solvable for eachf∈L0,ω, c∈R1.

2The boundary value problem4.1,4.5, where

lx≡xω−mxc, 4.5

and the norm of the linear functionalm:C_0,ω → R1_{is less than one is uniquely solvable} for eachf ∈L0,ω, c∈R1;

3The boundary value problem4.1,4.6, where

2k

j1 αjx

tj

c, 0≤t1< t2 <· · ·< t2k≤ω, 4.6

with0≤ −α2j−1 ≤α2j, j 1, . . . , k, and there exists an indexisuch that−α2i−1 < α2i,is uniquely solvable for eachf∈L_0,ω, c∈R1_.

4The boundary value problem4.1,4.7, where

2k

j1 tj

tj−1

αtxtdtc, 0t0≤t1< t2<· · ·< t2k≤ω, 4.7

in the case whenαt ≤ 0fort ∈ t2j−2, t2j−1, αt≥ 0 fort ∈t2j−1, t2j,

t2j

t2j−2αtdt ≥

0, j 1, . . . , k, and there existsj such thatt2j

t2j−2αtdt > 0,is uniquely solvable for each

Proof. If we suppose in the contrary that the assertions 1–4 are not true, then according to

Lemma 4.1, the nontrivial solutionx of homogeneous problem 4.3 exists. According to

Lemma 2.13see also Remark 2.14, the maximum boundaries principle is true, moreover, solutions of the homogeneous equationMx 0 does not decrease on0, ω.Conditions of each of the assertions 1–4 lead us to the inequalitylx /0 which contradicts to the existence of the nontrivial solutionxof the homogeneous problemMx0, lx0.

Theorem 4.3. Conditions of each of Theorems 2.1–2.8 imply assertions 1–4 of Theorem 4.2 for equations2.12,2.14,2.16,2.22,2.24,2.27and2.29respectively.

In order to proveTheorem 4.3we have only to note that conditions ofTheorem 4.2for corresponding equations follow from the conditions of each of Theorems2.1–2.8.

Remark 4.4. The condition

A1t≥B1t fort∈0, ω, 4.8

cannot be set instead of condition4.4as the following example demonstrates.

Example 4.5. Consider the boundary value problem

xt−x0 b1txh1t ft, t∈0,7,

x7−1

2x0 c,

4.9

where h1t and b1t are defined by formula 2.19 respectively. Here the operator A : C0,ω → L0,ωis defined asAxt x0and inequality4.8is fulfilled, but the operator As is a zero operator fors > 0,and inequality4.4is not true. Formula2.20defines the nontrivial solution of the homogeneous problem

xt−x0 b1txh1t 0, x7−1

2x0 0, t∈0,7. 4.10

According toLemma 4.1, problem4.9cannot be uniquely solvable for eachf ∈L0,ω, c∈ R1_.

Remark 4.6. The periodic problem

xt ft, t∈0, ω, xω−x0 c, 4.11

wherem 1,demonstrates that the conditionm < 1 in the assertion 2 ofTheorem 4.2

is essential: the functionxt ≡ 1 fort ∈ 0, ωis a nontrvial solution of the homogeneous boundary value problem

xt 0, t∈0, ω, xω−x0 0, 4.12

Remark 4.7. According toLemma 4.1, the problem

xt ft, t∈0, ω, 2k

j1 αjx

tj

c, 0≤t1< t2<· · ·< t2k≤ω, 4.13

demonstrates that the condition about existence of suchithat−α2i−1 < α2iin the assertion 3 is essential, and the problem

xt ft, t∈0, ω, 2k

j1

t2j

t2j−2

αtxtdtc, 0t0 ≤t1< t2<· · ·< t2k≤ω, 4.14

demonstrates that the condition about existence of suchithatt2j

t2j−2αtdt >0 in the assertion 4

ofTheorem 4.2is essential. If we suppose that suchidoes not exist, then the functionxt≡1 fort∈0, ωis a nontrvial solution of each of the homogeneous boundary value problems

xt 0, t∈0, ω, 2k j1αjx

tj

0, 0≤t1 < t2<· · ·< t2k≤ω,

xt 0, t∈0, ω, 2k

j1

_t2j

t2j−2

αtxtdt0, 0t0≤t1< t2<· · ·< t2k≤ω.

4.15

Remark 4.8. The conditionαt≤0 fort∈t2j−2, t2j−1in the assertion 4 ofTheorem 4.2cannot be omitted that follows from the example of one of the reviewers: the functionx1tis a nontrivial solution of the boundary value problem

xt x0, t∈0,2,

2

0

αtxtdt0, 4.16

whereαt 10 for 0≤t <1/2, αt −10 for 1/2≤t <1, αt 1 for 1≤t≤2.In this case t0 0, t1 1, t2 2,

2

0αtdt1 and consequently all other conditions of the assertion 4 of

Theorem 4.2are fulfilled.

Remark 4.9. Let us define the set

E{t∈0, ω:A1t>B1t}. 4.17

and the following condition:

athere exists a setEof nonzero measurei.e., mesE >0.

Remark 4.10. Instead of the condition about existence of suchithat−α2i−1 < α2iin the assertion 3 ofTheorem 4.2we can assume that

α2k>0, mes{0, t2k∩E}>0. 4.18

Condition4.18is essential as the following example demonstrates.

Example 4.11. The homogeneous boundary value problem

xt−x

t 2

b1tx

t 3

0, t∈0,2, 4.19

x1−x

1 2

0, 4.20

where

b1t

⎧ ⎪ ⎨ ⎪ ⎩

1, 0≤t≤1, 1

2, 1< t,

4.21

has a nontrivial solution

xt

⎧ ⎪ ⎨ ⎪ ⎩

1, 0≤t≤1, 1

2t1, 1< t.

4.22

In this case we havek1, t1 1/2, t21, E 1,2, mes{0,1∩E}0.

Remark 4.12. Let us denote asE1 the following set:E1{t:αt>0}.Instead of the condition about existence of suchithatt2j

t2j−2αtdt >0,in the assertion 4, we can assume the following:

mes{0, t2k∩E∩E1}>0.

Example 4.13. Consider 4.19, where the coefficient b1t is defined by 4.21, with the boundary condition

−

1

0

xtsin 2πtdt0. 4.23

Homogeneous problem4.19,4.23has a nontrivial solution defined by4.22. In this case we havek1, t10, t21, E 1,2, mes{0,1∩E∩E1}0.

Consider now3.1with the opposite sign near the neutral termSxt.

Proof follows from the fact that according to Lemma 3.8, solution x of the homo-geneous equation

xt Sxt−Axt Bxt 0, t∈0,∞. 4.24

does not decrease. Conditions of each of the assertions 1–4 of Theorem 4.2lead us to the maximum boundaries principle and consequently to the inequalitylx /0 which contradicts to the existence of the nontrivial solutionxof the homogeneous problemMx0, lx0.

Theorem 4.15. Conditions of each of Theorems 3.1–3.6 imply assertions 1–4 of Theorem 4.2 for equations3.3,3.5,3.8,3.11,3.14and3.17respectively.

Proof follows from the fact that conditions of each of Theorems3.1–3.6imply that the conditions ofLemma 3.8are fulfilled.

Note that Remarks4.4–4.12are relevant also for Theorems4.14–4.15.

Acknowlegments

This research was supported by The Israel Science Foundation Grant no. 828/07. The authors thank the reviewers for their valuable suggestions.

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